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Heat Flow and a Faster Algorithm to Compute the Surface Area of a Convex Body
 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
, 2006
"... We draw on the observation that the amount of heat diffusing outside of a heated body in a short period of time is proportional to its surface area, to design a simple algorithm for approximating the surface area of a convex body given by a membership oracle. Our method has a complexity of O(n4), wh ..."
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We draw on the observation that the amount of heat diffusing outside of a heated body in a short period of time is proportional to its surface area, to design a simple algorithm for approximating the surface area of a convex body given by a membership oracle. Our method has a complexity of O(n4), where n is the dimension, compared to O(n8) for the previous best algorithm. We show that our complexity cannot be improved given the current stateoftheart in volume estimation. 1
Randomized interior point methods for sampling and optimization
, 2009
"... We present a Markov chain (Dikin walk) for sampling from a convex body equipped with a selfconcordant barrier, whose mixing time from a “central point ” is strongly polynomial in the description of the convex set. The mixing time of this chain is invariant under affine transformations of the convex ..."
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We present a Markov chain (Dikin walk) for sampling from a convex body equipped with a selfconcordant barrier, whose mixing time from a “central point ” is strongly polynomial in the description of the convex set. The mixing time of this chain is invariant under affine transformations of the convex set, thus eliminating the need for first placing the body in an isotropic position. This strengthens and previous results of [11] for polytopes and generalizes these results to arbitrary convex sets. In the case of a convex set K defined by a semidefinite constraint of rank at most α and at most m additional linear constraints, our results specialize to the following statement. Let s ≥ pq  for any chord pq of K passing through a point x ∈ K. Then, after t = O n(m+ nα) n ln((m+ nα)s) + ln